16
Sampling
The sampling theorem, which is a relatively straightforward consequence of
the modulation theorem, is elegant in its simplicity. It basically states that a
bandlimited time function can be exactly reconstructed from equally spaced
samples provided that the sampling rate is sufficiently high-specifically, that
it is greater than twice the highest frequency present in the signal. A similar
result holds for both continuous time and discrete time. One of the important
consequences of the sampling theorem is that it provides a mechanism for exactly representing a bandlimited continuous-time signal by a sequence of
samples, that is, by a discrete-time signal. The reconstruction procedure consists of processing the impulse train of samples by an ideal lowpass filter.
Central to the sampling theorem is the assumption that the sampling frequency is greater than twice the highest frequency in the signal. The reconstructing lowpass filter will always generate a reconstruction consistent with
this constraint, even if the constraint was purposely or inadvertently violated
in the sampling process. Said another way, the reconstruction process will always generate a signal that is bandlimited to less than half the sampling frequency and that matches the given set of samples. If the original signal met
these constraints, the reconstructed signal will be identical to the original signal. On the other hand, if the conditions of the sampling theorem are violated,
then frequencies in the original signal above half the sampling frequency become reflected down to frequencies less than half the sampling frequency.
This distortion is commonly referred to as aliasing,a name suggestive of the
fact that higher frequencies (above half the sampling frequency) take on the
alias of lower frequencies.
The concept of aliasing is perhaps best understood in the context of simple sinusoidal signals. Given samples of a sinusoidal signal, many continuoustime sinusoids can be threaded through the samples. For example, if the samples were all of equal height, they could correspond to samples of a sinusoid
of zero frequency or in fact a sinusoid at any frequency that is an integer multiple of the sampling frequency. From the samples alone there is clearly no
way of determining which of the continuous sinusoids was sampled. The reconstruction filter, however, makes the assumption that the samples also correspond to a frequency less than half the sampling frequency; so for this particular example, the reconstructed output will be a constant. If, in fact, the
16-1
Signals and Systems
16-2
original signal was a sinusoid at the sampling frequency, then through the
sampling and reconstruction process we would say that a sinusoid at a frequency equal to the sampling frequency is aliased down to zero frequency
(DC).
Thus, as we demonstrate in this lecture, if we sample the output of a sinusoidal oscillator and then reconstruct with a lowpass filter, as the oscillator
frequency increases from zero, the output of the lowpass filter will correspondingly increase. The output frequency will match the input frequency until the oscillator frequency reaches half the sampling frequency. As the oscillator frequency continues to increase, the output of the lowpass filter will begin
to decrease in frequency.
It is important to understand that in sampling and reconstruction with an
ideal lowpass filter, the reconstructed output will not be equal to the original
input in the presence of aliasing, but samples of the reconstructed output will
always match the samples of the original signal. This relationship is emphasized in this lecture through a computer movie. It is also important to recognize that aliasing is not necessarily undesirable. As we illustrate with a hopefully enjoyable and entertaining visit with Dr. Harold Edgerton at MIT's
Strobe Laboratory, stroboscopy heavily exploits the concept of aliasing. Specifically, by using pulses of light, motion too fast for the eye to follow can be
aliased down to much lower frequencies. In this case, the strobe light represents the sampler, and the lowpass filtering is accomplished visually.
Suggested Reading
Section 8.0, Introduction, pages 513-514
Section 8.1, Representation of a Continuous-Time Signal by Its Samples: The
Sampling Theorem, pages 514 to mid-519
Section 8.3, The Effect of Undersampling: Aliasing, pages 527-531
Sampling
16-3
MARKERBOARD
16.1
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TRANSPARENCY
16.1
M
2w,
Spectra associated
with sampling a signal
based on amplitude
modulation with an
impulse train carrier.
M
o,
- WS
ci
(i
(W
W
2
- WM)
Signals and Systems
16-4
X(O)
TRANSPARENCY
16.2
Lowpass filter used for
recovery of the input
signal.
M
(M
P(W)
-2w,
-,
2o,
,
x (W)
H(w)
... A
A\1
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f
Wm
W
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(WS-WCM)
X(W)
TRANSPARENCY
16.3
Illustration of aliasing
when the sampling
frequency is too low.
P(W)
2i7frf*
- 2ws
- WS
2cws
Cis
co
Cos
s
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Sampling
16-5
p(t) =
6(t-nT)
n=- -00
x(t)
Xr (t)
WS,>
s
M
M
2
TRANSPARENCY
16.4
Block diagram of
sampling and
reconstruction using
an ideal lowpass filter.
wm
oS
H(w)
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-
K
L
-
LOM
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(Lis
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TRANSPARENCY
16.5
AI\
WKE NT
5 KC
Time waveforms and
spectra for a sampled
sinusoidal signal, with
no aliasing.
Signals and Systems
16-6
T
TRANSPARENCY
16.6
The same as
Transparency 16.5,
with the frequency
increasing.
LPF
KC
10 '
T
TRANSPARENCY
16.7
The input frequency
increased to the point
where aliasing occurs.
T
LPF
AL IASING
5KC
KC
5
10
10
Sampling
16-7
AThN
N
V
PF
/
4ILVJT
TRANSPARENCY
16.8
An input frequency
considerably above
that at which aliasing
occurs.
MY
ALIASING
MARKERBOARD
16.2
0 SC. lid.
v
H (J
F-7
Signals and Systems
16-8
DEMONSTRATION
16.1
Audio demonstration
of aliasing with a
sinusoidal oscillator.
TRANSPARENCY
16.9
Discrete-time
processing of
continuous-time
DISCRETE-TIME PROCESSING
OF CONTINUOUS-TIME SIGNALS
signals.
xConversion
x[n]
ds--c---r-t---im
Discrete-time
s
y[n]
Conversion to
coninuus-im
Sampling
16-9
TRANSPARENCY
16.10
Conversion of a
continuous-time signal
to a discrete-time
signal interpreted in
two steps. The
continuous-time signal
is first sampled with a
periodic impulse train,
and the impulse train
values are then
converted to a
discrete-time
sequence.
DEMONSTRATION
16.2
Use of a strobe to view
the motion of an
oscillating spring.
C/D conversion
p(t)
Conversion of
x (t)
Xx (t)
t
sr
sequence
ex[n]
=xe(nT)
Signals and Systems
16-10
DEMONSTRATION
16.3
Use of a strobe to view
the motion of a
rotating fan.
DEMONSTRATION
16.4
A rotating disk
observed with a
strobe.
I
Sampling
16-11
DEMONSTRATION
16.5
Water drops as seen
with a strobe.
I
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Resource: Signals and Systems
Professor Alan V. Oppenheim
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